LEDs are so efficient that even microamps can power them. If your LED driver is cheap, it'll run on basically nothing, or charge up enough to start for a fraction of a second.
The microamps come from a hot wire running next to a switched wire behaving as a capacitor when carrying AC voltage, letting microamps leak through. (It's not required that the light is on the hot side of the switch as I said previously, my bad).
This can happen if the switch box is a terminal box with hot and switched wires in the same cable, which is rather common. Probably some other configurations too.